Solve 2x+5/x-x/x-3x-3/x^2-x+3/left(x+1right)^2

$$\frac{ 2x+5 }{ x } - \frac{ x }{ x-3 } \frac{ x-3 }{ { x }^{ 2 } } - \frac{ x+3 }{ { \left(x+1 \right) }^{ 2 } }$$

Evaluate

$$\frac{2x^{3}+7x^{2}+7x+4}{x\left(x+1\right)^{2}}$$

Short Solution Steps

Multiply \frac{x}{x-3} times \frac{x-3}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x+5}{x}-\frac{x\left(x-3\right)}{\left(x-3\right)x^{2}}-\frac{x+3}{\left(x+1\right)^{2}}$$

Cancel out x\left(x-3\right) in both numerator and denominator.

$$\frac{2x+5}{x}-\frac{1}{x}-\frac{x+3}{\left(x+1\right)^{2}}$$

Since \frac{2x+5}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x+5-1}{x}-\frac{x+3}{\left(x+1\right)^{2}}$$

Combine like terms in 2x+5-1.

$$\frac{2x+4}{x}-\frac{x+3}{\left(x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and \left(x+1\right)^{2} is x\left(x+1\right)^{2}. Multiply \frac{2x+4}{x} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{x+3}{\left(x+1\right)^{2}} times \frac{x}{x}.

$$\frac{\left(2x+4\right)\left(x+1\right)^{2}}{x\left(x+1\right)^{2}}-\frac{\left(x+3\right)x}{x\left(x+1\right)^{2}}$$

Since \frac{\left(2x+4\right)\left(x+1\right)^{2}}{x\left(x+1\right)^{2}} and \frac{\left(x+3\right)x}{x\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x+4\right)\left(x+1\right)^{2}-\left(x+3\right)x}{x\left(x+1\right)^{2}}$$

Do the multiplications in \left(2x+4\right)\left(x+1\right)^{2}-\left(x+3\right)x.

$$\frac{2x^{3}+4x^{2}+2x+4x^{2}+8x+4-x^{2}-3x}{x\left(x+1\right)^{2}}$$

Combine like terms in 2x^{3}+4x^{2}+2x+4x^{2}+8x+4-x^{2}-3x.

$$\frac{2x^{3}+7x^{2}+7x+4}{x\left(x+1\right)^{2}}$$

Expand x\left(x+1\right)^{2}.

$$\frac{2x^{3}+7x^{2}+7x+4}{x^{3}+2x^{2}+x}$$

Expand

$$\frac{2x^{3}+7x^{2}+7x+4}{x\left(x+1\right)^{2}}$$

Short Solution Steps

Multiply \frac{x}{x-3} times \frac{x-3}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x+5}{x}-\frac{x\left(x-3\right)}{\left(x-3\right)x^{2}}-\frac{x+3}{\left(x+1\right)^{2}}$$

Cancel out x\left(x-3\right) in both numerator and denominator.

$$\frac{2x+5}{x}-\frac{1}{x}-\frac{x+3}{\left(x+1\right)^{2}}$$

Since \frac{2x+5}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x+5-1}{x}-\frac{x+3}{\left(x+1\right)^{2}}$$

Combine like terms in 2x+5-1.

$$\frac{2x+4}{x}-\frac{x+3}{\left(x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and \left(x+1\right)^{2} is x\left(x+1\right)^{2}. Multiply \frac{2x+4}{x} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{x+3}{\left(x+1\right)^{2}} times \frac{x}{x}.

$$\frac{\left(2x+4\right)\left(x+1\right)^{2}}{x\left(x+1\right)^{2}}-\frac{\left(x+3\right)x}{x\left(x+1\right)^{2}}$$

Since \frac{\left(2x+4\right)\left(x+1\right)^{2}}{x\left(x+1\right)^{2}} and \frac{\left(x+3\right)x}{x\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x+4\right)\left(x+1\right)^{2}-\left(x+3\right)x}{x\left(x+1\right)^{2}}$$

Do the multiplications in \left(2x+4\right)\left(x+1\right)^{2}-\left(x+3\right)x.

$$\frac{2x^{3}+4x^{2}+2x+4x^{2}+8x+4-x^{2}-3x}{x\left(x+1\right)^{2}}$$

Combine like terms in 2x^{3}+4x^{2}+2x+4x^{2}+8x+4-x^{2}-3x.

$$\frac{2x^{3}+7x^{2}+7x+4}{x\left(x+1\right)^{2}}$$

Expand x\left(x+1\right)^{2}.

$$\frac{2x^{3}+7x^{2}+7x+4}{x^{3}+2x^{2}+x}$$